In a classical annealer, the solution to a hard optimization problem is found by mapping the problem to a model system, and then simulating the model system on a conventional computer (i.e., a non-quantum computer such as a digital computer). In the simulation of the model system, the temperature of the model system is lowered, and as the temperature is lowered in the simulation, the energy of the model system drops into a low energy state that is a nearly optimal solution of the optimization problem.
Quantum annealers are hardware systems (e.g., a quantum computer such as the D-WAVE ONE, by D-Wave Systems) that find solutions to hard optimization problems by evolving a quantum system from a simple initial state to a solution of the hard problem by slowly changing the parameters of a control Hamiltonian. With a quantum annealer, unitary time evolution and quantum tunneling can provide speedup over classical annealing and classical optimization algorithms.
A quantum annealer is initialized with a quantum system at low or zero temperature with very strong quantum fluctuations and with the quantum system in a simple and known state. The quantum fluctuations allow the quantum system to tunnel between states. The strength of the quantum fluctuations are slowly turned off to arrive at the optimal or a near-optimal solution.
Concretely, at the initial time t=0 the quantum system of the quantum annealer is initialized in the ground state (i.e., lowest energy state). The quantum system is described by a Hamiltonian H(0) that evolves over time (i.e., the Hamiltonian is a function of time H(t)) to a final Hamiltonian H(tfinal) at time t=tfinal. The low energy states of the Hamiltonian H(tfinal) provide good solutions to an optimization problem. Tunneling effects due to quantum mechanics give such a quantum annealer a substantial advantage over classical annealers.
However, it is very expensive and hard to build quantum hardware for a quantum annealer.
Further, simulating quantum systems is exponentially hard which more than wipes out any advantage due to quantum tunneling.